Sunday, January 31, 2016

Below are the questions with answers and explanations for Part 3 and 4 of the Algebra I (Common Core) Regents exam for January 2016. The multiple-choice questions appeared in a previous post. Part II questions appeared in a another post.

As always, in order to get this thread up quickly, the images have been omitted. They will be added at a later date.

Each question in Part 3 is worth 4 credits, for a total of 16 credits. Partial credit will be given. The one question in Part 4 is worth 6 credits.

January 2016 Algebra 1 (Common Core) Regents, Part 3

33. Let h(t) = -16t2 + 64t + 80 represent the height of an object above the ground after t seconds. Determine the number of seconds it takes to achieve its maximum height. Justify your answer.

State the time interval, in seconds, during which the height of the object decreases. Explain your reasoning.

The maximum height is at the vertex, and the vertex is on the axis of symmetry.
The axis of symmetry is x = -b/(2a) = -64/(2(-16)) = -64/-32 = 2 seconds.
h(2) = -16(2)2 + 64(2) + 80 = -16(4) + 128 + 80 = -64 + 128 + 80 = 144. It will be 144 feet off the ground.

The object is descending from 2 &lt t &lt 5.
At 5 seconds, it is no longer descending. Before 2, it was still rising.

34. Fred's teacher gave the class the quadratic function f(x) = 4x2 + 16x + 9.
a) State two different methods Fred could use to solve the equation f(x) = 0.
b) Using one of the methods stated in part a, solve f(x) = 0 for x, to the nearest tenth.

a) Pick any two methods for solving are factoring, completing the square and the quadratic formula. There is also graphing, but that won't be helpful if the answer is not an integer.

Note that factoring may not be possible. If it turns out that it isn't, cross that one out and use the other two.
4x2 + 16x + 9 = 0 -- 4 and 9 are the squares of 2 and 3 and (2)(2)(3) = 12, not 16, so it is (2x + 3)2. (One could hope it would be easy.)

Personally, I prefer the Quadratic formula is Completing the Square is going to involve fractions anyway, so look at the illustration:

So the solutions are {-3.3, -.7}

35. Erica, the manager at Stellarbeans, collected data on the daily high temperature and revenue from coffee sales. Data from nine days this past fall are shown in the table below. (image omitted)

State the linear regression function, f(t) that estimates the day's coffee sales with a high temperature of t. Round all values to the nearest integer.

State the correlation coefficient, r, of the data to the nearest hundredth. Does r indicate a strong linear relationship between the variables? Explain your reasoning.

You need to put the data into LISTs in the graphing calculator and then do a Linear Regression. Before the exam, your calculator should have had its memory reset. After that, the two things that should have happened were that it was put back into Degree Mode, and DiagnosticsON should have been executed. It's a function in the CATALOG. With that on, the correlation coefficient will appear on the screen when you run a Linear Regression.

Put the temperatures in L1, and the sales in L2. Double check your work. (I had one typo, and that would have skewed my answer!)
Hit Stat, go to the Calc menu, and press 4: Linear Regression. Press ENTER.

f(t) = -58t + 6182 -- use f(t) and t. Don't use y and x.

The correlation coefficient is -.94 (to the nearest hundredth). This indicates a strong negative relationship because the number is close to -1, meaning that it is almost a straight line.

36.A contractor has 48 meters of fencing that he is going to use as the perimeter of a rectangular garden. The length of one side of the garden is represented by x, and the area of the garden is 108 square meters.

January 2016 Algebra 1 (Common Core) Regents, Part 4

37.The Reel Good Cinema is conducting a mathematical study. In its theater, there are 200 seats. Adult tickets cost $12.50 and child tickets cost $6.25. The cinema's goal is to sell at least $1500 worth of tickets for the theater.

Write a system of linear inequalities that can be used to find the possible combinations of adult tickets, x, and child tickets, y, that would satisfy the cinema's goals.

Graph the solution to this system of inequalities on the set of axes on the next page. Label the solution S.

Marta claims that selling 30 adult tickets and 80 child tickets will result in meeting the cinema's goal. Explain whether she is correct or incorrect, based on the graph drawn.

The graph will be coming shortly. Please be patient.

The system of inequalities is:

x + y &lt 20012.50x + 6.25y >1500

When you graph them, both will have solid lines. The first inequality (# of tickets) will be shaded below. The second inequality (money from the sales) will be shaded above the line. The area shaded twice will be get the S.

Marta is incorrect. (If you graphed correctly) If you look at the graph, (30, 80) is not in the section with the S, so it is not a solution to the system of inequalities. (You needed to refer to the graph, so just plugging the numbers into both inequalities is not sufficient.)

Saturday, January 30, 2016

Below are the questions with answers and explanations for Part 2 of the Algebra I (Common Core) Regents exam for January 2016. The multiple-choice questions appeared in a previous post. The rest of the questions will appear in a later post.

As always, in order to get this thread up quickly, the images have been omitted. They will be added at a later date.

Each question in Part 2 is worth 2 credits, for a total of 16 credits. Partial credit will be given. Basically, you can have one computational, conceptual, graphing or rounding error, but as long as you have a consistent answer, you can still get a point. Two different mistakes, and there is no credit for the answer.

You could have also completed the square or used the quadratic formula after putting it in standard form. If you made one computational error, but continued until the end and gave an answer, you would've gotten one point.

28.The graph below shows the variation in the average temperature of Earth's surface from 1950-2000, according to one source. (image omitted)
During which years did the temperature variation change the most per until time? Explain how you determined your answer.

The largest change was between 1960 and 1965 when the slope of the graph was -.15/5. It is the steepest part of the graph. The increase from 1975 to 2000 is a constant .1/5.

Be careful with the fractions and decimals. I just typed them incorrectly, but I caught the mistake before I posted them. (Had the decimal point in the wrong position.)

29.The cost of belonging to a gym can be modeled by C(m) = 50m + 79.50, where C(m) is the total cost for m months of membership.

State the meaning of the slope and y-intercept of this function with respect to the costs associated with the gym membership.

It costs $79.50 to joint the gym. That is a one-time fee that you pay even if you go for zero months. $50 is the monthly fee, which is paid for the number of months, m.

Note that this was just a definition question with nothing to work out. Common Core is doing a lot of that.

30. A statistics class surveryed some students during one lunch period to obtain opinions about television programming preferences. The results of the survey are summarized in the table below. (image omitted)
Based on the sample, predict how many of the schools 351 males would prefer comedy. Justify your answer.

Because a > b, that makes (b - a) a negative number, and when you divide by a negative number, the inequality symbol has to flip around.
(Also, since a > b, (b - a) cannot equal zero, so it is okay to divide by it.)

32. Jacob and Jessica are studying the spread of dandelions. Jacob discovers that the growth over t weeks can be defined by the funtion f(t) = (8)*2t. Jessica finds that the growth function over t weeks is g(t) = 2t+3.

Calculate the number of dandelions that Jacob and Jessica will each have after 5 weeks.

Based on the growth from both function, explain the relationship between f(t) and g(t).

f(t) = 8(2)5 = 8(32) = 256.
g(t) = 25+3 = 28 = 256.

Based on the growth, the two functions are the same.
This is because g(t) = 2t+3 = 2t * 23 = 2t * 8 = f(t).

I'll be honest here. I have no clue what they are going for in this last question. Based on only one data point, you can only conclude that the are the same function for that one input. It isn't enough to say the functions are the same. It is easy to prove that they are the same (as I showed above) but that isn't what they asked.

Friday, January 29, 2016

Below are the questions with answers and explanations for Part 1 of the Algebra I (Common Core) Regents exam for January 2016, the multiple-choice questions. The open-ended questions will be posted separately.

As always, in order to get this thread up quickly, the images have been omitted. They will be added at a later date.

Each question in Part 1 is worth 2 credits, for a total of 48 credits. Usually, a score of 30 credits on the entire test is worth a score of 65. The curve is steep after that. To achieve a final score of, say, 75, you will need roughly 55 credits. (The exact curve will not be revealed for a few days.)

January 2016 Algebra 1 (Common Core) Regents, Part 1

(2) 2. The vertex is at the point (h, k) taken from the general form, f(x) = (x - h)2 + k.

2. The graph below was created by an employee at a gas station. (image omitted)
Which statement can be justified by using the graph?

(2) For every gallon of gas purchased, $3.75 was paid. (1) and (3) can be eliminated by checking the graph. Neither (10, 35) nor ((2, 5) are points on the line. You can't see (1, 3.75) but if you multiply by 4, you will see (4, 15) on the graph. Choice (4) is just ridiculous -- the graph has nothing to do with miles driven.

3. For a recently released movie, the function y = 119.67(0.61)x models the revenue earned, y, in millions of dollars each week, x, for several weeks after its release.

Based on the equation, how much more money, in millions of dollars, was earened in revenue for week 3 than for week 5?

(3) 17.06. Substituting 3 for x in the function gives us about 27.16 million, and substitution 5 gives us 10.11, with a difference of 17.05. The difference with the answer is a minor rounding error.

(1) II only. The first is the sum of two rationals, which is rational. The second is the sum of a rational and an irrational, which is irrational. The third squares the square root of 5, which is 5, a rational number. The last is three times the square root of 49, which equals 3 * 7 which is 21, a rational number.

5. Which inequality is represented by the graph below?

(2) y > 2x - 3. The y-intercept is -3. The slope is 2. The graph is shaded above the line. (The line is also solid, but that doesn't matter for the choices given.)

6. Michael borrows money from his uncle, who is charging him simple interest using the formula I = Prt. To figure out what the interest rate, r, is, Michael rearranges the formula to find r. His new formula is r equals

(3) I/Pt. Divide both sides of the equation by P and t to isolate r.

7. Which equation is equivalent to y - 34 = x(x - 12)?

(4) y = (x - 6)2 + 2.
Distribute the x and you get: y - 34 = x2 - 12x.
Add 34 to both sides: y = x2 - 12x + 34.
There are no integer factors of 34 that have a sum of -12. (Eliminate choices 1 and 2.) This means completing the square, OR working backward from the other choices.
Squaring (-6) gives us +36. To make 36 into 34, we need to subtract 2. Choice (4).

8. The equation A = 1300(1.02)7 is being used to calculate the amount of money in a savings account. What does 1.02 represent in this equation?

(4) 2% growth. It's greater than 1, so it is growth. (And if it is a savings account, it better be growth, or why have the account?) The decimal .02 is 2% as a percentage.

11. Joe has a rectangular patio that measures 10 feet by 12 feet. He wants to increase the area by 50% and plans to increase each dimension by equal lengths, x. Which equation could be used to determine x?

13. The table below (image omitted) shows the cost of mailing a postcard in different years. During which time interval did the cost increase at the greatest average rate?

(4) 2006-2012. Find the average rate for each interval. From 1898-1971, the change was 5/73. From 1971-1985, it was 8/14. From 1985-2006, it was 10/21. From 2006-2012, it was 11/6. Choice (4) is the only one greater than 1, so it is obviously the greatest.

If you eliminate choice 1 as obviously very flat, you can sketch out the other points and see that 2006-2012 is the steepest line.

14. When solving the equation x2 - 8x - 7 = 0 by completing the sqaure, which equation is a step in the process?

(2) (x - 4)2 = 23. Half of 8 is 4, so eliminate choices (3) and (4). (-4)2 = +16. You have to add 16 to both sides AND add 7 to both sides to get the 7 to the other side of the equation. 16 + 7 = 23.

15. A construction company uses the function f(p), where p is the number of of people working on a project, to model the amount of money it spends to complete a project. A reasonable domain for this function would be

(1) positive integers. The number of people has to be a counting number. It cannot be negative nor a fraction. People can also be zero, but in the context of the problem, if zero people are working on a project, the company won't make any money. (None of the options include zero.)

16. Which function is shown in the table below? (image omitted)

(4) f(x) = 3x. Substitute 0 into the functions and only choice (4) works. For that matter, choice (4) is the only one that can produce fractions as output when the input is integers.

(3) f(1) = 3, f(n + 1) = 2f(n) + 1. Each term after the first is one more than twice the previous term.

Note: I don't know if there are typos in choices (1) and (2) or if it was intended to write f(n) as an exponent. It's just odd-looking. As an exponent or not, the answers are incorrect, so it doesn't matter.

19. The range of the function defined as y = 5x is

(2) y > 0. A positive number to any exponent will be positive, never zero or negative.

(1). The graph will move 1 space to the left and 2 down because h = -1 and k = -2.

21. Which pair of equations could not be used to solve the following equations for x and y?

4x + 2y = 22-2x + 2y = -8

(4) 8x + 4y = 44; -8x + 8y = -8. In choices (1) - (3), one or both of the equations is multiplied by some constant. In (4), the -8 was not multiplied by 4 but the left side of the equation was.

22. The graph representing a function is shown below. (image omitted)
Which function has a minimum that is less than the one shown in the graph?

The graph has a minimum at (3, -7). This eliminates choices (2) (-3, -6) and (4) (8, 2).
In choice (1), the axis of symmetry is 3, and the y-coordinate of the vertex is y = (3)2 - 6(3) + 7 = 9 - 18 + 7 = -2
In choice (3), the axis of symmetry is 1, and the y-coordinate of the vertex is y = (1)2 - 2(1) - 10 = 1 - 2 - 10 = -11.
Alternatively, you could have put these into your graphing calculator and just observed the correct answer.

23. Grisham is considering the three situations below.
I. For the first 28 days, a sunflower grows at a rate of 3.5 cm per day.
II. The value of a car depreciates at a rate of 15% per year after it is purchased.
III. The amount of bacteria in a culture triples every two days during an experiment.
Which of the statements describe a situation with an equal difference over an equal interval?

(1) I, only. The first situation is a linear function, growing the same amount every day. (Constant slope.) The other two are exponential functions with the amount changing from interval to interval, even if the percentage remains the same.

24. After performing analyses on a set of data, Jackie examined the scatter plot of the residual values for each analysis. Which scatter plot indicates the best linear fit for the data? (images omitted)

(3). The residual plot should contain randomly-scattered points above and below the x-axis. It should not have a pattern to it. Choices (1), (2) and (4) show curve-like patterns to the plotting of their residuals. These indicate a poor fit for the data.

I easily had over 50 of these, but they were a bit repetitive after a while, which is why I added the years to the equations.
If this is popular, then, as with all good movies and many bad ones, there will be a sequel.

The Answer Key will be up soon -- as soon as I find it. Yes, a couple of them are stumping me now and I wrote the thing!

Thursday, January 28, 2016

It's that time of year again: Regents time. I will be uploading parts of the Regents exams for Geometry and Algebra over the next few days. Please, be patient -- I have to type all this in by hand. And that doesn't count the amount of stuff that I need to scan in.

As always, the images will be scanned in when available, and the entries will be updated.

New York Geometry (Common Core) Part 2

25.Triangle ABC is graphed on the set of axes below. (image omitted) Graph and label triangle A’B’C’, the image of triangle ABC after a reflection over the line x = 1.

There are two parts to this: you need to both graph it and label it to get both of them.
The line x = 1 is the vertical line one unit to the right of the y-axis. So your answer is a triangle with vertices A’ (5, 0), B’(2, 4) and C’(2,0).
The vertices need to be labeled correctly, or you could probably get away with writing the labels and coordinates below the grid.
If you forgot something, you lose a point. If you do some other transformation, including a reflection over a different line, you will lose half-credit, which is one point.

Because DC is parallel to BA with BC as a transversal, then the alternate interior angles are congruent. So &ltDCB = &ltABC = 30 degrees.
Because OB and OA are both radii, OB = OA, so triangle OAB is an isosceles triangle, with base angles equal to 30 degrees. 30 + 30 = 60. 180 – 60 = 120.
m &ltAOB = 120o

If you solved this using inscribed angles and arcs, more power to you. If you got the correct answer, you will get full credit.

27. Directed line segment PT has endpoints whose coordinates are P(-2, 1) and T(4, 7). Determine the coordinates of point J that divides the segment in the ratio 2 to 1. The use of the set of axes below is optional.

The difference of the x values is 6. The difference in the y values is also 6. Two thirds of 6 is 4.
Add 4 to the x and y value of P. J(-2 + 4, 1 + 4) = J(2, 5)

28. As graphed on the set of axes below (image omitted), triangle A’B’C’ is the image of triangle ABC after a sequence of transformations.

Is triangle A’B’C’ congruent to triangle ABC? Use the properties of rigid motions to explain your answer.

Triangle A’B’C’ is the image of ABC reflected over the y-axis and also translated down 3 units. Reflections and translations are rigid motions that do not affect the size or shape, so the image is congruent.

29. A carpenter leans an extension ladder against a house to reach the bottom of a window 30 feet above the ground. As shown in the diagram below, the ladder makes a 70o angle with the ground. To the nearest foot, determine and state the length of the ladder.

This used to be an Algebra topic, but now is in Geometry, involving Trigonometry ratios.

The ladder and the wall form a right triangle. You know the bottom angle. You have the length of the opposite side. You need the length of the hypotenuse. Opposite and Hypotenuse means using Sine.

You will lose a point if you use the incorrect ratio, for making an incorrect calculation, or not rounding correctly. Zero if you make more than one mistake.

You will not be penalized twice for a consistent error.

30. During an experiment, the same type of bacteria is grown in two petri dishes. Petri dish A has a diameter of 51 mm and has approximately 40,000 bacteria after 1 hour. Petri dish B has a diameter of 75 mm and has approximately 72,000 bacteria after 1 hour.

(image omitted)

Determine and state which petri dish has the greater population density of bacteria at the end of the first hour.

The population density is the population divided by the Area. You need to find the area of each circle and then divide the population by that area.

Don’t forget to divide the diameters by 2 to get the radii.

AA = (pi)(25.5)2 = 2042.82…
Divide 40000 by 2042.82 = 19.58

AB = (pi)(37.5)2 = 4417.86…
Divide 72000 by 4417.86 = 16.297

Petri dish A has the greater population density.

Important: Normally, you shouldn’t round in the middle of a problem. I only did it here because a) I didn’t need an exact answer, and b) I have to divide by the number that I rounded. I could have written the equations to avoid that, but then I wouldn’t have found the individual areas. Not that they were needed in this problem.

31. Line L is mapped onto line M by a dilation centered at the origin with a scale factor of 2. The equation of line L is 3x – y = 4. Determine and state an equation for line M.

This is a surprisingly simple question to answer, but a little more complicated to explain. The slope of the line will not change. The x-intercept and y-intercepts will double (be twice as far away from the origin).

So the answer is obviously3x – y = 8 or some variation.

But how do you show it?

You could rewrite the equations in slope intercept form, and then double the y-intercept: So 3x – y = 4 becomes 3x – 4 = y, with the answer of y = 3x – 8.

32. The aspect ratio (the ratio of screen width to height) of a rectangular flat-screen television I s16:9. The length of the diagonal of the screen is the television’s screen size. Determine and state, to the nearest inch, the screen size (diagonal) of this flat-screen television with a screen height of 20.6 inches.

This can be solved using ratios and Pythagorean Theorem, or by using Trigonometric Ratios.

Tuesday, January 26, 2016

I will post the questions and answers as soon as I can HOWEVER, no one in my current school was scheduled to take the Geometry Regents today, so I could not obtain a copy. I will have one by this Friday, at the latest, because I will be grading that exam over the weekend up in Williamsburg, Brooklyn.

I will definitely have a copy of the Algebra exam on the day it is given.

As for Algebra 2, which I don't usually post, I don't believe anyone will be taking the exam this month. Maybe in June.

Sunday, January 24, 2016

My Week in Geek

It was a good week for the geek in me last week. There was plenty to watch and catch up on, and most of what I saw did not disappoint.

The week started off in retro fashion when I discovered that one of the episodes of Doctor Who on my DVR was actually a Tom Baker serial, The Seeds of Doom. The DVR recorded it a few months ago when BBC America ran a few "Breakfast with Baker" specials on Sunday mornings. I hadn't realized that it was there.

Excellent series, and I recommend tracking it down. It starts off as if it might be a riff on the original The Thing when alien plant life is discovered in the Antarctic. (Yes, that's the other end of the Earth from the movie, but a pole's a pole!) However, the action moves back to England, and it gets to be more Day of the Triffids, but it's the human villain that's creepy as hell as he sides with the plants to take over the world and eradicate the Animal kingdom, of which is no longer seems to relate to.

Also peculiar in this serial is that the TARDIS is absent until the very end, after the story has concluded.

Next up was Face-Off, the Syfy reality series, a competition where effects designers compete to make unusual make-ups for actors in a short turnaround, usually three days. A lot is asked of them, and most times, most of them deliver. Apparently, these challenges aren't too far from what actual effects experts (including the judges) can be called on to do by a director with a very quick deadline. I generally sum this show up as saying that it's Project Runway, with science fiction makeup and costumes. Because it is. It's the same format, except I can actually sit through this, without being forced to. It's my favorite show on TV because I'm amazed at what these competitors bring to the final reveal stage.

The amusing thing this week is that I recalled that after the first episode, I tweeted, "no one whose face I want to slap" because sometimes there's that contestant that just grates you (or the producers go out of their way to frame someone that way), but not so far. This week, in social media, I discovered that the Germans have a word, backpfeifengesicht, which means "a face that’s begging to be slapped." I needed to learn its pronunciation.

Add to this that my favorite person on the show, so far, is the German guy, not because of his talent (so far) but because I love listening to him speak. Partly, this is because he reminds me of some old sitcom characters. I'm thinking more Get Smart than Hogan's Heroes, and not Siegfried, either.

Moving on ...

Agent Carter had a two-hour premiere, which was more like two one-hour episodes, but that didn't matter much. They move Peggy Carter to the West Coast for a special assignment, and she's right back in the thick of things with Jarvis. The season will undoubtedly deal with "zero matter" which behaves in a funny way, like the obelisk in Agents of SHIELD. Also back is Peggy's old nemesis, the Russian spy who grew up as part of the Black Widow program. Things should get good.

Arrow and Flash had good episodes, moving their plotlines along, setting up the emergence of new characters. I don't want to spoil anything that might be coming but Felicity in a wheelchair seems as obvious as the introduction of Wally West. (Likewise, someone gave me some speculation on Diggle, but it referred to a character from a time when I wasn't really paying much attention to comics.) Not that any of this is a bad thing.

Which brings me to the biggie: DC's Legends of Tomorrow, which looks to be a new anthology series of sorts. A group of "B-characters" from Arrow and Flash are given new life on this show. I will not join in on suggestions that each of them could hold a series on their own (mostly because I don't believe that) but they should be able to make a heck of a team. The set-up also provides the producers with a way to replace unpopular characters or actors who wish to leave.

Central to the plot is the one new character: Time-traveling Rip Hunter, played by Arthur Darvill who has some experience in the matter from his seasons of Doctor Who. I won't say he was wasted as Rory, but one episode of Legends tells me he was underused. (Plus, I saw him on Broadway "once".)

Of the other characters, the new Firestorm will take some getting used to. I read the comic from its beginning until around the time DC decided that the "Nuclear Man" should really be a Fire Elemental. They seem to have ditched that idea. He appeared in one episode of the cartoon Brave and the Bold, an incarnation I didn't like. However, Prof. Stein has had plenty of screen time on Flash and is a good character, and the actor plays the part well. Jax needs to hold up his end.

As for the villains who may be heroes, they should fit in because they are a little too four-color even for Flash, but in this show, over-the-top should work fine. Heatwave needs to develop a little more personality, like Cold has.

Rounding out the week with more reruns...

Friday brought another airing of Galaxy Quest, which I have dubbed the Third Greatest Star Trek Movie Ever. Seeing Alan Rickman one more time brought smiles and laughs, which is better at the time than seeing Die Hard would've been.

And finally, I've been recording episodes of Quantum Leap from an oldies network on cable. I loved the show, but missed many of the episodes because it moved around the schedule a lot back then. Who remembers what it might've been opposite. One of the first episodes I taped was the "Man of La Mancha" episode, where Sam leaps into an understudy for the musical, playing in Syracuse, and he sees the understudy for Dulcinea -- his former piano teacher that was his first crush/love, and who was formerly linked up with the person Sam is inhabiting. There are a few reasons why this is a favorite of mine, but watching it mine added another: the Guest Stars. They could get actual Broadway people for this episode. John Cullum appeared in 1776 a few years before this and played the actor playing Don Quixote. Also appearing, but not singing, was Ernie Sabella also of Broadway, TV and The Lion King. There had a lot of great characters back then who appeared on a lot of shows like this one. There are still a lot of good actors today, but there are so many channels of TV that the talent pool is stretched way too thin.

So that was the week, and it was a good one. I'm hoping that weeks to come don't disappoint, but it will be hard to live up to this one.

Wednesday, January 20, 2016

Television was the center of family life ... until it was so successful that we had televisions in every room, and the family split up and retreated into their own areas. But still, the center was the living room TV because it was the greatest receiver of all, at least in terms of size. But then that stopped mattering.

For a long time, television sets were limited by the size of the cathode ray tube, making the set to heavy to move and too big to place on any piece of furniture. After a certain size, they became impractical.

And then the flatscreen arrived, and screen sizes (as measured by the diagonals) took off again.

But then something odd happened. Mini-TVs started to have more clarity, and then they slowly grew in size. And then streaming video on phones and tablets became practical, and, again, the clarity and the sharpness of the screens rivaled the bigger cousins.

Shortly after the turn of the 21st century, PDAs shrank and phones grew (after originally shrinking) until they became the same gadget. But then the gadgets kept growing, coincidentally at the same time that streaming has taken off. So people have given up 50 inch screens for 5 inch screens, but the miniatures are getting bigger again.

Personally, I have a big screen in the bedroom where my DVR is, but I'll occasionally watch shows from the DVR on my iPad, but that's for convenience, not a preference.

Monday, January 18, 2016

Welcome. If you found this page while looking for answers to Common Core Algebra Regents answers, please read what this column has to say.
Or just review my Regents tab for more answers and reviews.

I am a New York City high school teacher and I have been grading Regents exams for about 15 years. I have put in overtime grading the exams and have gone to training about what is acceptable and what isn't (and what might be worth, say, 1 point instead of 0). The scorers have very little leeway in how they grade things, but one thing is certain: you cannot get full credit for any problem without a completely correct response with the work shown or graphs labeled.

Most of what is written below may seem to be common sense. My students tell me that it is, and yet they make these little mistakes time and again.

Scoring Higher on the Common Core Algebra Regents

It's simple really. There are two things that you have to do:

Don't make silly mistakes

Don't give away points by skipping questions

One piece of advice you need to forget: Just write something. No, do not just write something. Write something thoughtful. If you take two numbers and multiply them, most times, it will be incorrect. If you use the Quadratic Formula but it isn't a quadratic equation, it is "totally incorrect".

Now the positive things you should do:

Round Your Decimals Correctly

Rounding errors will cost you one point. For a two-point question, that's half the credit, regardless of how much work you did.

If the problem says to round to the nearest tenth, then you include one decimal place. If it says nearest hundredth, two decimal places, etc.

If the problem is money, then you have either two place or zero places (nearest cent or nearest dollar). Never write $2.5 as your answer. (I've seen it.)

If the next digit is 5-9, round up. If the next digit is 0-4, round down.

And DO NOT confuse TENTH with TEN and HUNDREDTH with HUNDRED. (Again, I've seen it.)

Final warning: DO NOT round in the middle of the problem. Wait until the end to round your numbers or errors will creep in.

If you round 1/3 to .3 in the middle of the problem and then multiply (.3)(60), you will get 18 instead of the correct answer 20. You just lost a point.

Use the Correct Formulas -- and Write Them Out

Some of the formulas are given to you, but not all of them. Don't expect the one you need to be there. Don't use one just because it is there. (Seriously, if you're in high school you should know that the Area of a rectangle is length times width and the Volume of a rectangular prism is length times width times height. I shouldn't see pi mentioned.)

Write the formula out. It isn't required, but it can't hurt. And it will help when you substitute for the variables that you have.

Calculator Issues: Use Parentheses

Calculators have a bunch of issues. Make sure you know how to use them and how to find the functions that you think you might need.

In particular, don't forget to use parentheses. Some of the biggest mistake involve parentheses.

Square Roots: SQRT(4 + 9 is NOT the same as SQRT(4) + 9.
I can't draw the root symbol, but the calculator will give you an open parenthesis. It doesn't require a close parenthesis. However, the syntax of your equation may DEMAND one.

Absolute Value: abs(-8 + 3 - 4 is NOT the same as abs(-8 + 3) - 4.
Same reasoning as with Square Roots. The close of the Absolute Value in the expression means close the parentheses in the calculator.

Don't Forget to Answer the Question

If you do all that work to figure out what x equals, make sure you go back and answer the question. Was x in the question or did you make it up (use it for the unknown in the problem)? Did the question ask you to find x, or did it ask to find some other expression that uses x?

If you found x, do you also need to find y, and state the answer as coordinates?

Whatever they ask, answer it. Don't stop and say, well, I did enough. "It's obvious." "It means the same thing." Why take a chance?

Answer in a Sentence

Write out your answer. Put it in a box. If you clearly defined your variables, you have half of a sentence already. Just put an = sign and the answer.

Make sure that your final solution can be found, especially if you have a lot of work on the paper. (Also, "a lot of work on the paper" is A Good Thing.)

Label Your Graphs

Along with labeling your answers to algebraic solutions, you need to label your graphs. If there are two equations or inequalities, you must label at least one of them. (Both would be better.) Again, it is NOT "obvious" which one is which. The scorer needs to know that you know. So tell them.

Furthermore, if it is a system of equations, label the point of intersection. If the lines do not meet at an integer point, you probably made a mistake. HOWEVER, if you cannot find a mistake that you made, estimate the answer as best as you can. You can get a point for a "consistent mistake" if you correctly label the intersection of two lines, even if one is incorrectly graphed.)

If you label a bunch of points when you graph -- e.g., (1, 3), (2, 5), (3, 7) ... -- then make sure you circle the solution to the system and label it as such. You won't get credit for having (3, 7) written if there are a bunch of other points labelled as well.

Speaking of Graphs -- DON'T SKIP THEM

Some students are intimidated by the graphs? Why? You have been doing them for a good part of the year. And you should have a graphing calculator at your disposal.

Using a calculator means that you may have to rewrite an equation to isolate y, so you can use the y= key.

Once you GRAPH it, look at the TABLE. You have all the points you need to plot the graph.

And, speaking of which, DRAW THE GRAPH FROM EDGE TO EDGE. Don't plot four points, connect the dots and stop.

The One Exception to this rule is if you are given a domain to use, such as, -4 &lt x &lt 4, in which case, DO NOT GO PAST THOSE NUMBERS and do NOT use ARROWS.

And, obviously, you won't be able to go to an edge in the numbers are outside of the domain of the function, such as with the square root function.

For inequalities, shade one inequality with lines going in one direction. Shade the other with lines going in a different direction. The criss-cross pattern is your solution. Put a big "S" in that section of the graph. Don't just scribble in three sections of the graph.

Look Below the Graph for Extra Questions

If there is a question below the graph, ANSWER IT. Don't ignore it. Don't say you didn't see it.

If they are for a point that is in the solution to a system of inequalities, remember that any point on a broken line, including the intersection point, is NOT in the solution.

EXPLAIN

If you are asked to "explain", then that's what you need to do. And you need to reference what is on the page and what mathematical concepts or principles are being addressed.

Take it from me, whatever B.S. you might put on your English essays will not work here. By the way, it doesn't work in English class, either.

Don't write, "I don't agree with Angelica's solution because she is wrong." That isn't an explanation to why Angelica isn't correct.

Trust Yourself and Succeed

Be careful. Look over your work with an eye for these "silly" mistakes. I say "silly" because students don't like when I say "STUPID" in class. They think I'm calling them "stupid" instead of the mistake.

A second read-through can find mistakes you missed the first time.

Follow these tips and you'll score a few extra points that you would've lost otherwise. Every point matters. You won't know what mistake will drop you below 65 or 75 or 90, or whatever your target grade is.

Just do your best, but remember that cutting corners and skipping questions is NOT doing your best.

Monday, January 11, 2016

Yes, it's that logical dilemma for any math-oriented person. Knowing that the odds of winning are ridiculous, and what you're expected return is -- especially, if you're in a pool with 19 other people!

And yet some people who wouldn't do these things will still take trips to Las Vegas or Atlantic City or any of the Indian casinos. Sure, they'll say, it's about the shows, the experience, the entertainment value! Yeah, and you still plunk down hard cash at a table or a one-armed bandit (which don't even have any arms any more!).

It comes down to this: someone will eventually win. And if, somehow, it turns out to be someone you know, you don't want to be the one who didn't even think of buying a ticket. As rational as the square root of 3, to be sure, but just because it's irrational doesn't make it any less real.

Saturday, January 02, 2016

Well, my first victory is to acknowledge to myself that I am, in fact, an educator. Sometimes I forget that. Sometimes I remember that but it doesn't really seem like that. Sometimes it seems like outside forces are acknowledging it only to try to counter that reality.

No, I know I am a Teacher, and that isn't going to change any time soon.

But plans? What's the expression about "a battle plan is only good until . . . " No, I do not see my students, any of them, as "the enemy", even if some are openly hostile to me or to learning in general. But many grand plans -- especially those coming from boring PD -- seem to crash and burn when they hit the classroom. Maybe the resources weren't there or in working order. Maybe the backup wasn't what it should've been. Maybe the response wasn't what was expected.

And for every plan, you need 30 (individual) contingencies and another few backups, in case of snafus.

(Not the way I thought I'd start the blog in the New Year, but then I didn't think I'd end the old year by throwing out my back, helping someone, so bear with me.)

I admire the people I follow online who can make grand plans. For a better part of the last couple of years, I assured them I would model some of their suggestions should I ever have a permanent classroom again. (I've been floating around the system for a couple of years.) Well, I finally got a classroom, and the challenges I face are substantial -- and unmatched by any of my current online peers. Should I look for new Twitter chats? I doubt the teachers facing what I do have time for them.

As it is, I've cut back on the one or two education-based chats that used to participate in weekly -- back when I didn't have papers to grade, parents to call, anecdotals to write, online systems to update. Email is a wonderful tool -- I wish the parents used it in connection with their child's educational welfare.

I plan One Day at a Time. Whenever I plan too far ahead, I have to alter those plans and most of the effort has been wasted.

I might have a goal, but I've seen so many plans crash and burn that I've been sticking to things that work. Try something new? Sure, I'll try to sneak things in, and gauge the reaction and effectiveness.

But first I have to get over the feeling that I'd prefer to "float around the system" again. I don't hate being in the classroom. But I'm close to saying I hate being in some classrooms.

Okay, end of rant. I'll try to be more positive tomorrow. Or maybe next week.

About Me

Mr. Burke is a high school math teacher in New York as well as a part-time writer, and a fan of science-fiction/fantasy books and films.
He started making his own math webcomic totally by accident as a way of amusing his students and trying to make them think just a little bit more.
Unless otherwise stated, all math cartoons and other images on this webpage are the creation and property of Mr. Chris Burke and cannot be reused without permission.
Thank you.